= Centered four-cycle identity for a biregular graph
{title2=$t_4(G)=\delta^4+n^{-4}\operatorname{tr}((HH^{\mathsf T})^2)$}
For an equal-part <biregular graph> with part size $n$ and degree $\delta n$, let $M$ be its <biadjacency matrix> and $H=M-\delta J$, where $J$ is the all-ones <matrix>. The constant row and column sums make $H$ annihilate constants on both sides. Since $MM^{\mathsf T}=\delta^2nJ+HH^{\mathsf T}$ and the two summands have zero products, their squared <matrix trace> splits as displayed. The <bipartite four-cycle density> counts the squared <matrix trace> of $MM^{\mathsf T}$ divided by $n^4$.
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